This work presents a fully integrated system for reliable grasping and manipulation using dense visual mapping, collision-free motion planning, and shared autonomy. The particular motion sequences are composed automatically based on high-level objectives provided by a human operator, with continuous scene monitoring during execution automatically detecting and adapting to dynamic changes of the environment. The system is able to automatically recover from a variety of disturbances and fall back to the operator if stuck or if it cannot otherwise guarantee safety. Furthermore, the operator can take control at any time and then resume autonomous operation. Our system is flexible to be adapted to new robotic systems, and we demonstrate our work on two real-world platforms - fixed and floating base - in shared workspace scenarios. To the best of our knowledge, this work is also the first to employ the inverse Dynamic Reachability Map for real-time, optimized mobile base positioning to maximize workspace manipulability reducing cycle time and increasing planning and autonomy robustness.